The mapping class group is the group of isotopy classes of orientation preserving homeomorphisms of .

Assume complexity is at least 2 (otherwise, MCG is virtually free).

We shall use various notions of rank.

1. Version 1

Let be a Hadamard manifold. Assume has a discrete cocompact isometry group . The quasiflat rank is the maximal dimension of quasiflats, i.e. quasi-isometrically embedded Euclidean spaces. Replacing quasi-isometrically by isometrically does not change the definition (this is a theorem of Bruce Kleiner).

However, for non-uniform lattices, the story is still open. Robert Young solved the case of .

Theorem 5 (Behrstock-Drutu) For , .

For , .

In small genera, we can improve the second estimate.

Difficulties: cutting spheres in pieces.

3. Version 3

Let be a Hadamard manifold. Let us define higher dimensional divergence: it measures how geodesics diverge, and simultaneously, filling properties.

Fix a base point . Let be a sphere of volume volume, outside a large ball. Define

Example 2 symmetric space. Then

If , .

If , (Leuzinger).

If , (Hindawi), improved by Wenger into .

Theorem 6 (Behrstock-Drutu)

If , .

If , .

Again, in low genera, one can improve a bit. Also, 1-dimensional divergence is better understood.

Theorem 7 (Abrams-Brady-Duchin-Young)

4. Intermediate results

Here are two facts that hold in a larger generality.

If a group is and has a combing, then

If is hyperbolic, then all are linear (for Lipschitz maps, this is due to Urs Lang).

5. Proof

I explain why MCG’s do not behave like Hadamard spaces.

For all QFlatRank , there exists a -quasflat that is maximal, i.e. not contained in a quasi-flat of higher dimension. This is different from the case of symmetric spaces. These lower dimensional quasiflats arise from multicurve splittings of the surface.

For instance, split where is an annulus. Then Dehn twists in provide undistorted subgroups in MCG.

Our main tool: let be a subsurface. Then there is a Lipschitz map of to which is locally constant outside .